Mathematical Foundations of Quantum Mechanics

454 Pages·1955·9.675 MB·english

by  Neumann, J. von.

MATHEMATICAL
FOUNDATIONS
. OF
QUANTMUEMC HANICS
ByJ OHNv oNN EUMANN.
.
tranfsrlotamhtG eee dr meadnib tyi on
ROBET.RB TE YER
-,lP.RlNCETON
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PRINCETON UNIVERSITY PRESS
. .
1955
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.·. Copyright, by PrinceUtnoni versPirteys s
1955,
LondonG:e offrCeuym berlegOex,f orUdn iversPirteys s
L.CC.o rd
53-10143
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Printiendt heU ntiedS tateosfA merica
TRANSLATOPRR'EFSA CE
Thitsr anstliaofno llcolweosls yt htee xotft he
oringailG ermaend it.i Tohnet arnslatmeaudnsc irphta s
·
ben� caerfullyr evdi bsyet he thaousrot hatth ied eas
expsresedin thivso luamrehe i rsa trh tehant hoosfet he
tarnslatanodr a,n E.ro�y gs fro�mh o� riaglint xeta re
alsdou et ot haeu trh.o· '!-:i.:.,. :
Thet ranslawtiosrh ees xprteohs iss deep
grat:f:tu_Pdr�o��f's�s.vo�orC?n Nef'umoahrnni sv eyr consider-
.
..a ble r ..,..... ther enderoifn-g�i hdee oasf't hoer ingail
efforts�in
voluimneta o t r�lsatiwohinc who ulcdo nvtehyes ame
meanings.
RobeTr.tB eyer
ProvciedRe,n .I .
Decbeemr19,49

PREF'ACE
Theo bjecotft hibso oiks t op reenstt hnee w
quantumm echcasnin i aun ifireedp resenwthiacthsi,oo n
fara si ti sp osbslianed eufsu,li sm a�heimclaalty
riguosr.o This qnuanetwum m echacnshi asi nr enctey ears
achieivnie dte ss nstieapl arwthsa its p reumasblay
definiftoirm:v et hes oc-all"etdr a.ns::,·af�to::r,:'rni,ot,nh� e� o.ry"
Thereftohpreer ipnaclei mh�� ra1b1e p lacoentd h e
� �!�
�
generanda flu namdentqaule i':i.on'shwichha vaer isienn
\
cnnoecitowni tthh itshe or.y Inp atrilcau,rt hdei fficult
prombsl eionft erpretmaatyni oofwn hi,c ahr ee venno nw ot
fullrye solwvieldl, i nvbeeis gtatiendd e ta.i ltIhni s
cnotextthre elatoifqo unna tumm echacnsit os tatistics
andt ot hec lsaiscasltt aiisctamlec hacnsii so fs pieacl
imporntcea.H owevewres, h aallsa r uel omainytd iscus­
.
sioonft haepp licaotfiq ounant umm echianclam ethotdos
parctuilparrbo lmes, waesl als adnyi ssciuosonf s pieacl
theordieersi fvreodtm h ge enertahle o--rayt l eassot
fara st hiissp osbsliwei thoeudntan geritnhunge d erstand­
ingo ft hgee nerraell atipos.n shTihsiemsest hmeo re
advibslasei nsceee vraelx cel.tlreenatnttsmo eft hese
1
prloebmasr ee ithienpr r inotri np roceosfsp ubilcat.i on
1
Theraer eam,o nogt rhse,t hfeo lwlioncgo mprehensive
tremaetnst:S ommerfSeulpldpe,m etnott he4h t editoifo n
Atmobauun d Sperkatlli,nB iruneanscwheig19,2 ;8W eylT,h e
vii
viii PREFACE
Ont heo thhearn da,p rensteatoifot nh mea the­
maticaln etcoseoaslrfsyo trh peurp oseosft hitsh eJo r"
.
wilbleg ivein.,,e ..taheyo roHfi lebrsta pcaen dthe
soc-allHeedrm iteoapnae torr.s Forth iesn ,d accurate
an
introdutcout nbioounn doepde raitsoa rlssn oe scaesry,i ..e,
ane xtenosfit ohnet heyo bryeonidt csl saiscalli mits
(edvelobpye dH ilanbdeE r.Ht e llgienrF,-R ieszE,.
Schdmti, Toeptlz)i,T hef olowlimnagy b es airdea grding
o.
thmee theopmdl oyiendt himso doef t rtemae:n atsa lreu,
calculasthiooubnlespd e rfowrmietdth h oep eorarttsh em­
selv(ewsh ircehps rneet ypschiaqlu antiets)ani dn otw ith
thmea tircewshi,c ha,f tethrine t orductoifoa n ( pseacli
anda briatryr)c ordoiantes ysmt ienH ilebrtsp caer,e sult
frotmh e.m Thi"soc oraditenf' ree", i e..·i, nvar,im aentthod,
witiht sst rgolnyg eoemtrliacn ggue·ap, osseesssn oteiacble
f'ormaald. vanst.ag e
Diarci,n s eveprpaaelr, s waesl alsi nh is
2
renctelpybu lisbhoeodk h,asgi vearn epresentation of'
quantmume chcansiw hich siacsre clyt o sbuearp ssde in
breyvan idte legean,ac ndw hicihsa tt hes amet mieo f
invaricanhtaa rct.e·I rti st herepfeohrrasepf itntgti o
advanac fee wa rgumensto nb ehaolffo umre th_wohid,c h
deviactoenssr iadbelfyr otmh �to fD iarc.
Thme ethoofDd i arc,m entioabnoevde( ,and this
iso vleoroktedoda yi na g repaatr otfq uantmume chanical
TheooryfG rouapnsdQ unatumM echiancs(r t.b yH .P .R obert­
sno)L,o dnon19, 13.;F renkle,W aevM echiancsO,xf or·d1 9,23;
Borannd JordEanl,ee mntaQruanet enmcehan,iB kerl·i1 9n03,;
Diarc,T heP rinciopfQl ueanst umM echiancs2,n de d., .
Oxford,19 63.
2
Cf.P orc.R o.yS o.cL ondo10n9, 1 9(52)an d thfeo lolwing
isuse,se spieaclly.!.(.!]_91 2)6.I ndedpeenntolfDy i rca,
p. Jordanz,. P yhsi�k (192)6an d F.L odnonz., P yhski 4o
(912)6g ave similafro undiaotnfso trh teh eyo.r
PREFACE ix
lietrarte,ub eucsaoef t hcel araintdey el gnacoef t he
theo)ir nyn ow ays atistfhireee sq uirse moemfna ttheim­at
carli rg o--noetv einft heasreer educiena dn atural
and
propfears hitootn h eex tecnmmoto ne lwsheerien t heoreti­
caplhy scis. Forex pamlet,h mee thaodhde rteost he
ficttihoanet a cshe ldfj-oaionpte ractaobnre p uti n
diganoalf orm. Int hec asoeft hoospee ratowrhsci hf or
thiissn ota ctualtlhyce asteh,i rse quitrheiesn troduc­
tioonf" impropfeWlrc"t iwointssh e lfn-tacrdoictor.y
propesr.tT iheein sretioofns ucahm athaetmilc "aifciton"
isf renqutyel necesrys ianD iarc'aspr poa,ce hvetnh ough
the praotbh lanedim s m erelyo fco ancleu liantg rniucmael­
ly the orfea s culalertld ye fineexepdr mien.t Thewroeu ld
.
ben oo bejcitohne rieft hecsoen cse,wp hticchan nobte
incorpeodri antto ptrhenest de ayf ramewoorfank a ylsis,
werinet rnisiclalny ecersyfs oatrh pehy silc taheo.ry
Thu,.sa sN ewtomneicannh iacfsi rbsortu ghtab outt he
develoopmfet nhtie n fitneisaimlc allcuu,sw hicihni, ts
.originfaolrm w,a Wlsd obutdelny ots elf-connt,ss io·s te
quantmume chiancmsi ghts uggesat nsetwru ucrtef ooru r
"analysiosf i nfininia.ntye vlayr bilae"s- -i..e,t hmea hte­
mati�taelci hnquweo uhladv teob ec anhgedan,d n ott he
physilc.hateo.r Buyt thibsy imnseo a ns csate.h eI t
shourltadh ebrep o:I;ntouettdh atth qeu natumm echanical
"rTansfaotrmiothne ocranyb "e e stahbeldiin s a emran n
whicihsj usatsc leaanrdW lifibeudtw, h icihsa lsow ith­
oumta theimclaa otbjencst.i Iots hodu _lbee mpshiaztehda t
thce orrects tcrutunreee d cnoonts ini stam athaetmilc a
refineanmde enxtp laantioofnt hDei ramce thboudtr, a ther
.
thaittr equiarp ersod cuerdei ffefrriontmgh vee rybe­
ginning, namelyt,hre e liaonntc heHe i lebrtth eooryf
operrast.o
In thanea lsyiosf t hfeun damentquaelsi to,ni st
wilbles hownt hhseo twa tciasflto iurmlaosfq u antum
mechacnsic abn ed erifvreodam f ewq uailtiavteb.a sic
assumptioFnusrh.te rmo,rt ehewriel ble a d etaidliesd-
x PREFACE
cussoino ft hper oblaestm o w hteheirti sp ossbilte o
tratches tatiicsactlh aarctoefrq uantumm echcasnt ioa n
e
ambiguitiy.. e,i ncpolmeentsesino ur desicprtono f
( ) �
natu.r Iendeeds,u cahni netrpreitoawnto ubldea .natural
conmciotaonftt hgee ne·rparlinctihpaeltae c phr obability
E�ci ta rsiefsr om itnohcmepe ltenoefso su knro wle.d ge
lt� �
Thiesx panlatio"nbhy i ddpeanamr ete,r"ass w elals
anotherre,l attoei dt, w hicahsc irbetsh e" hidden
paa.mretetro"t hoebs revearn dn ott ot hoeb sersvyesdt em,
has bperpeoons meorde t hano nce. Howeervi,t waipplela r
thatth icsan s cracelsyu ccienea d s ataicstfowrya, yo r
morper ecisseulcayhn,e xplaniaotins i nocmpiabtle with
cearitqnu ailtatfiunvdeam entpaolts uleasot fq uantum
3
mechcas..ni
Ther elatoifto hnse es ttasititcost hermo­
dynamicsi sa lsoc onsid.e rcAel dosinevret sigatsihoonw s
thatth wee l·kln ownd ifflitciuoefsc lsaiscmechanisc,
�
whicahr ree lattoet dh e" doirsedra"ss umptionsc·enasersy
fotrh feo undatiooftn h rmeodyniamcs,c anb ee liminated
4
her.e
3
Cf.C h. and Ch.VI .3.
IV.
4
er.c h.
v.
C ONT E NT S
Page
TRAN SLATORS' PREFACE v
PREFACE vii
CHAPTER I .
IntroudctoCryo nsiderations
1 TheO rigino f't heT ransf'ormatTihoeno ry 3
•
2.T heO ringailF ormualtioonfsQuan tum Mechacnsi 6
3.T heEq uivaelncoef t heT woT heioer:s
TheT ransformaTthieoonry 1 7
4.T heE quiavlencoeft h�T woT heorsi:e
HilebrtS pace 28
CHAPTER II
AbstraHcitl beSrpta ce
1 TheDe f'iniitono f'H ilberStp ace 34
•
2. TheG eometorfHy i lebrtS pace 46
3.D irgesisono nt heC ondiitonsA .- E. 59
4.C lods LeineaMrain f'olds 73
5.Op eraotrsi nH ilebrtS pace 87
6 TheE igenvalue Problem 120
•
7.C onitnuaiton 170
8 .In itiaClo nisderiaotnCso ncerning the
· EigenvaPlruoeb lem· 119
9.D igrsesoino f't heEx istneceand Uniqusensoe f'
the Sioolnuostf' t heE igenvaPlruoeb lem 154
1.0 CommuattivOep eroarts 107
1 1 TheT race· 187
•.
CHAPTER III
TheQuan tum Sttaistics
·1. TheS ttaisctailA sesrtioonfsQ uaumnt Mcehacnis· 916
2.T heS ttaisticla Inetrpretation 206
3.S iulmtaneouMse asubrlalyi andt Measubriialty
inG eneral 211
4.U necratintRye lations 230
5; Prjoectionass P rospiotniso 247
6 RadiaitonT heory 254
•
CHAPTER IV
DeducitveDe velmoepnto f' tThheeo r-1
• The Fundamental Boaf'ts hiesS tatiicslat Theory2 95
1
2. Proofo ft heS taitstilc Faormulas 313
3. ConclusFiroonExpmse rimesn t 328
xi
xii CONTENTS
V
CHAPTER
GenerCaoln isderations
1 .M easurmeentand Revebrisliity 347
2. ThermodyniamcalC onseirdaitons 358
3. Reversiitbyai nldEq uilibriPtunrb ole�s 379
4. TheM acrsocopiMce ausrement 398
CHAPTER
VI
TheM easuringP roecss
1. Formulatioofnt heP roblem 417
2. ComposiStyset mes 422
3. Dicsussiono ft heM easuringP roecss 437